Logarithmic potentials on $ {\mathbb{P}}^{n}$

Fatima Zahra Assila

doi:10.1016/j.crma.2018.02.004

Complex analysis

Logarithmic potentials on

P^{n}

Presented by: Jean-Pierre Demailly

Fatima Zahra Assila ¹

¹ Université Ibn Tofail, avenue de l'Université, Kénitra, Maroc

Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 283-287.

Abstract
Résumé

We study the projective logarithmic potential $G_{μ}$ of a probability measure μ on the complex projective space $P^{n}$ . We prove that the range of the operator $μ ⟶ G_{μ}$ is contained in the (local) domain of definition of the complex Monge–Ampère operator acting on the class of quasi-plurisubharmonic functions on $P^{n}$ with respect to the Fubini–Study metric. Moreover, when the measure μ has no atom, we show that the complex Monge–Ampère measure of its logarithmic potential is an absolutely continuous measure with respect to the Fubini–Study volume form on $P^{n}$ .

On étudie le potentiel logarithmique projectif $G_{μ}$ d'une mesure de probabilité μ sur l'espace projectif complexe $P^{n}$ . On établit que l'image de l'opérateur $μ ⟶ G_{μ}$ est contenue dans le domaine de définition (local) de l'opérateur de Monge–Ampère complexe agissant sur les fonctions quasi-plurisousharmoniques dans $P^{n}$ par rapport à la métrique de Fubini–Study. Si μ n'a pas d'atomes, on montre que la mesure de Monge–Ampère complexe du potentiel logarithmique de μ est absolument continue par rapport à la forme volume de Fubini–Study de $P^{n}$ .

Received: 2017-06-23
Accepted: 2018-02-12
Published online: 2018-02-21

DOI: 10.1016/j.crma.2018.02.004

Author's affiliations:

Fatima Zahra Assila ¹

¹ Université Ibn Tofail, avenue de l'Université, Kénitra, Maroc

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     title = {Logarithmic potentials on $ {\mathbb{P}}^{n}$},
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     pages = {283--287},
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     volume = {356},
     number = {3},
     year = {2018},
     doi = {10.1016/j.crma.2018.02.004},
     language = {en},
}

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Fatima Zahra Assila. Logarithmic potentials on $ {\mathbb{P}}^{n}$. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 283-287. doi : 10.1016/j.crma.2018.02.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.02.004/

References
Cited by

[1] E. Bedford; B.A. Taylor The Dirichlet problem for the complex Monge–Ampère equation, Invent. Math., Volume 37 (1976) no. 1, pp. 1-44

[2] Z. Blocki On the definition of the Monge–Ampère operator in $C^{2}$ , Math. Ann., Volume 328 (2004) no. 3, pp. 415-423

[3] Z. Blocki The domain of definition of the complex Monge–Ampère operator, Amer. J. Math., Volume 128 (2006) no. 2, pp. 519-530

[4] S. Bouckson; P. Essidieux; V. Guedj; A. Zeriahi Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262

[5] M. Carlehed Potentials in pluripotential theory, Ann. Fac. Sci. Toulouse Math. (6), Volume 8 (1999) no. 3, pp. 439-469

[6] E. Cartan Leçons sur la géometrie projective complexe, Gauthier-Villars, 1950

[7] U. Cegrell The general definition of the complex Monge–Ampère operator, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 1, pp. 159-179

[8] D. Coman; V. Guedj; A. Zeriahi Domains of definition of Monge–Ampère operator on compact Kähler manifolds, Math. Z., Volume 259 (2008) no. 2, pp. 393-418

[9] J.-P. Demailly Monge–Ampère operators, Lelong numbers and intersection theory (V. Ancona; A. Silva, eds.), Complex Analysis and Geometry, The University Series in Mathematics, Plenum Press, New York, 1993

[10] H. Federer Geometric Measure Theory, Springer Verlag, 1996

[11] S. Helgason The Radon transform on Euclidean spaces, two point homogeneous spaces and Grassmann manifolds, Acta Math., Volume 113 (1965), pp. 153-180

[12] V. Guedj; A. Zeriahi The weighted Monge–Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007) no. 2, pp. 442-482

[13] V. Guedj; A. Zeriahi Degenerate Complex Monge–Ampère Equations, EMS Tracts Math, vol. 26, 2017

[14] D.L. Ragozin Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., Volume 162 (1971), pp. 157-170

[15] E. Study Kürzeste Wege im komplexen Gebiet, Math. Ann., Volume 60 (1905), pp. 321-378

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